Vortex filament clustering in 3D Ginzburg-Landau theory

Ever since the seminal work of Bethuel-Brezis-Helein(BBH) on a simplified 2d Ginzburg-Landau system, a very active field in nonlinear analysis and mathematical physics that connects the theory of harmonic maps, energy renormalization and hard-analysis concentration estimates has been developed. However, up to this date, the beautiful description of vortex configurations achieved in the 2d setting is yet to find an exact analog in higher dimensions; this is due to a weaker characterization of vortex defects and the delicate interplay between the geometry of these codimension 2. In this talk I will present a framework that allows to capture an "effective interaction energy" of nearly parallel vortex filaments in certain 3d settings: this provides a next order or "renormalization" of the energy of clustering-filaments configurations together with a more accurate description of the vortex region in the spirit of the BBH study in 2d. This is joint work with Robert Jerrard.